On non-unique solutions in mean field games

Bruce Hajek, Michael Livesay

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

The theory of mean field games is a tool to understand noncooperative dynamic stochastic games with a large number of players. Much of the theory has evolved under conditions ensuring uniqueness of the mean field game Nash equilibrium. However, in some situations, typically involving symmetry breaking, non-uniqueness of solutions is an essential feature. To investigate the nature of non-unique solutions, this paper focuses on the technically simple setting where players have one of two states, with continuous time dynamics, and the game is symmetric in the players, and players are restricted to using Markov strategies. All the mean field game Nash equilibria are identified for a symmetric follow the crowd game. Such equilibria correspond to symmetric ϵ-Nash Markov equilibria for N players with ϵ converging to zero as N goes to infinity.In contrast to the mean field game, there is a unique Nash equilibrium for finite N. It is shown that fluid limits arising from the Nash equilibria for finite N as N goes to infinity are mean field game Nash equilibria, and evidence is given supporting the conjecture that such limits, among all mean field game Nash equilibria, are the ones that are stable fixed points of the mean field best response mapping.

Original languageEnglish (US)
Title of host publication2019 IEEE 58th Conference on Decision and Control, CDC 2019
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages1219-1224
Number of pages6
ISBN (Electronic)9781728113982
DOIs
StatePublished - Dec 2019
Event58th IEEE Conference on Decision and Control, CDC 2019 - Nice, France
Duration: Dec 11 2019Dec 13 2019

Publication series

NameProceedings of the IEEE Conference on Decision and Control
Volume2019-December
ISSN (Print)0743-1546

Conference

Conference58th IEEE Conference on Decision and Control, CDC 2019
CountryFrance
CityNice
Period12/11/1912/13/19

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization

Fingerprint Dive into the research topics of 'On non-unique solutions in mean field games'. Together they form a unique fingerprint.

  • Cite this

    Hajek, B., & Livesay, M. (2019). On non-unique solutions in mean field games. In 2019 IEEE 58th Conference on Decision and Control, CDC 2019 (pp. 1219-1224). [9029906] (Proceedings of the IEEE Conference on Decision and Control; Vol. 2019-December). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/CDC40024.2019.9029906